For the reduction and calculation of multiple polylogarithms there exist publicly available
tools [1]}, [2]}.
Some explicit examples for the shuffle sums are
\(S_{n,1,1}(x,y)&=\sum _{\vec{z}\in x\sqcup \mathchoice{\hspace{-4.44443pt}}{\hspace{-4.44443pt}}{\hspace{-1.94443pt}}{\hspace{-1.94443pt}}\sqcup y} \mathrm {Li}_{1,n+1}\left(\frac{z_2}{z_1},z_1\right)=\mathrm {Li}_{1,n+1}\left(\frac{y}{x},x\right)+\mathrm {Li}_{1,n+1}\left(\frac{x}{y},y\right)\,,\\S_{n,2,1}(x,y)&=\sum _{\vec{z}\in \vec{x}_2\sqcup \mathchoice{\hspace{-4.44443pt}}{\hspace{-4.44443pt}}{\hspace{-1.94443pt}}{\hspace{-1.94443pt}}\sqcup y}\mathrm {Li}_{1,1,n+1}\left(\frac{z_3}{z_2},\frac{z_2}{z_1},z_1\right)\nonumber \\&=\mathrm {Li}_{1,1,n+1}\left(\frac{y}{x},1,x\right)+\mathrm {Li}_{1,1,n+1}\left(\frac{x}{y},\frac{y}{x},x\right)+\mathrm {Li}_{1,1,n+1}\left(1,\frac{x}{y},y\right)\,,\\\\S_{n,2,2}(x,y)&=\sum _{\vec{z}\in \vec{x}_2\sqcup \mathchoice{\hspace{-4.44443pt}}{\hspace{-4.44443pt}}{\hspace{-1.94443pt}}{\hspace{-1.94443pt}}\sqcup \vec{y}_2} \mathrm {Li}_{1,1,1,n+1}\left(\frac{z_4}{z_3},\frac{z_3}{z_2},\frac{z_2}{z_1},z_1\right)\nonumber \\&=\mathrm {Li}_{1,1,1,n+1}\left(1,\frac{y}{x},1,x\right)+\mathrm {Li}_{1,1,1,n+1}\left(\frac{y}{x},\frac{x}{y},\frac{y}{x},x\right)\nonumber \\&+\mathrm {Li}_{1,1,1,n+1}\left(\frac{x}{y},1,\frac{y}{x},1,x\right)+\mathrm {Li}_{1,1,1,n+1}\left(\frac{y}{x},1,\frac{x}{y},y\right)\nonumber \\&+\mathrm {Li}_{1,1,1,n+1}\left(\frac{x}{y},\frac{y}{x},\frac{x}{y},y\right)+\mathrm {Li}_{1,1,1,n+1}\left(1,\frac{x}{y},1,y\right)\,.\) 
